How can I convert paths into one plane
Start by deselecting everything. Then move your cursor over a piece of your logo and press L this will select all connected vertices to the piece under your cursor. Now you can extrude E the selected section to the height you want.

Repeat this process for each piece. After extruding a section I would then hide it by selecting all linked vertices (the extrude will deselect the original vertices as you move the extruded side) and pressing H. When you are finished &#9095; AltH will un-hide the vertices.